#147206

order	512
nilpotency class	4
isoclinism	isoclinic class of #13156
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1936
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#10881
prime radical	C₂⁴	#137
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

487 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2)(3,14,18,41)(4,21)(5,28)(6,31)(7,13)(8,16,35,19)(9,24)(10,29)(11,32)(12,43)(15,48,51,64)(17,47,20,46)(22,45,54,62)(23,38)(25,44)(26,40)(27,39)(30,42)(33,57,61,58)(34,53,36,52)(37,50,63,55)(49,56)(59,60);(1,3,5,19,6,20,29,52,7,18,30,16,32,17,43,53)(2,8,12,14,11,36,42,47,13,35,10,41,31,34,28,46)(4,22,25,57,26,51,59,55,27,54,23,58,24,15,49,50)(9,37,38,64,39,61,60,45,40,63,44,48,21,33,56,62);(1,4)(2,9)(3,15,18,51)(5,23)(6,26)(7,27)(8,33,35,61)(10,38)(11,39)(12,44)(13,40)(14,45,41,62)(16,50,19,55)(17,54,20,22)(21,31)(24,32)(25,30)(28,60)(29,49)(34,63,36,37)(42,56)(43,59)(46,64,47,48)(52,58,53,57)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,d^2=h,e^2=i,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,h^a=hi,c^b=ch,d^b=di,e^b=ei,f^b=fi,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54402);